The subject of this paper is the rigorous derivation of a quasistatic evolution model for a linearly elastic{\textendash}perfectly plastic thin plate. As the thickness of the plate tends to zero, we prove via Γ-convergence techniques that solutions to the three-dimensional quasistatic evolution problem of Prandtl{\textendash}Reuss elastoplasticity converge to a quasistatic evolution of a suitable reduced model. In this limiting model the admissible displacements are of Kirchhoff{\textendash}Love type and the stretching and bending components of the stress are coupled through a plastic flow rule. Some equivalent formulations of the limiting problem in rate form are derived, together with some two-dimensional characterizations for suitable choices of the data.

}, keywords = {-convergence, Perfect plasticity, Prandtl{\textendash}Reuss plasticity, Quasistatic evolution, Rate-independent processes, Thin plates}, issn = {0294-1449}, doi = {https://doi.org/10.1016/j.anihpc.2012.11.001}, url = {http://www.sciencedirect.com/science/article/pii/S0294144912001035}, author = {Elisa Davoli and Maria Giovanna Mora} } @article {2012, title = {Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density}, number = {arXiv:0901.4041;}, year = {2012}, note = {21 pages}, publisher = {Elsevier}, abstract = {The asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied, as the thickness $h$ of the plate goes to zero. More precisely, it is shown that critical points of the nonlinear elastic functional $\mathcal E^h$, whose energies (per unit thickness) are bounded by $Ch^4$, converge to critical points of the $\Gamma$-limit of $h^{-4}\mathcal E^h$. This is proved under the physical assumption that the energy density $W(F)$ blows up as $\det F\to0$.

}, doi = {10.1016/j.jde.2011.09.009}, url = {http://hdl.handle.net/1963/3466}, author = {Maria Giovanna Mora and Lucia Scardia} } @article {2012, title = {Nonlinear thin-walled beams with a rectangular cross-section-Part I}, journal = {Math. Models Methods Appl. Sci. 22, 1150016 (2012)}, number = {SISSA;79/2010/M}, year = {2012}, publisher = {World Scientific}, abstract = {Our aim is to rigorously derive a hierarchy of one-dimensional models for thin-walled beams with rectangular cross-section, starting from three-dimensional nonlinear elasticity. The different limit models are distinguished by the different scaling of the elastic energy and of the ratio between the sides of the cross-section. In this paper we report the first part of our results.}, doi = {10.1142/S0218202511500163}, url = {http://hdl.handle.net/1963/4104}, author = {Lorenzo Freddi and Maria Giovanna Mora and Roberto Paroni} } @article {2011, title = {Quasistatic evolution in non-associative plasticity - the cap models}, journal = {SIAM Journal on Mathematical Analysis 44, nr. 1 (2012) 245-292}, number = {SISSA;05/2011/M}, year = {2012}, publisher = {SIAM}, abstract = {Non-associative elasto-plasticity is the working model of plasticity for soil and rocks mechanics. Yet, it is usually viewed as non-variational. In this work, we prove a contrario the existence of a variational evolution for such a model under a natural capping assumption on the hydrostatic stresses and a less natural mollification of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled.}, keywords = {Elasto-plasticity}, doi = {10.1137/110823511}, url = {http://hdl.handle.net/1963/4139}, author = {Jean-Francois Babadjian and Gilles A. Francfort and Maria Giovanna Mora} } @article {2011, title = {Large Time Existence for Thin Vibrating Plates}, journal = {Communication in Partial Differential Equations 36 (2011) 2062-2102}, number = {SISSA;59/2009/M}, year = {2011}, publisher = {Taylor \& Francis}, abstract = {We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large\\r\\ntimes under appropriate scaling of the initial values such that the limit system as h --> 0 is either the nonlinear von Karman plate equation or the linear fourth order Germain-Lagrange equation. In the case of the\\r\\nlinear Germain-Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation.}, doi = {10.1080/03605302.2011.618209}, url = {http://hdl.handle.net/1963/3755}, author = {Helmut Abels and Maria Giovanna Mora and Stefan M{\"u}ller} } @article {2011, title = {The matching property of infinitesimal isometries on elliptic surfaces and elasticity on thin shells}, journal = {Archive for Rational Mechanics and Analysis 200 (2011) 1023-1050}, number = {SISSA;75/2008/M}, year = {2011}, publisher = {Springer}, abstract = {Using the notion of Γ-convergence, we discuss the limiting behavior of the three-dimensional nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like h β with 2 < β < 4. We establish that, for the given scaling regime, the limiting theory reduces to linear pure bending. Two major ingredients of the proofs are the density of smooth infinitesimal isometries in the space of W 2,2 first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.}, doi = {10.1007/s00205-010-0387-6}, url = {http://hdl.handle.net/1963/3392}, author = {Marta Lewicka and Maria Giovanna Mora and Mohammad Reza Pakzad} } @article {2011, title = {Nonlinear thin-walled beams with a rectangular cross-section - Part II}, number = {SISSA;14/2011/M}, year = {2011}, institution = {SISSA}, abstract = {In this paper we report the second part of our results concerning the rigorous derivation of a hierarchy of one-dimensional models for thin-walled beams with rectangular cross-section..}, keywords = {Thin-walled cross-section beams}, url = {http://hdl.handle.net/1963/4169}, author = {Lorenzo Freddi and Maria Giovanna Mora and Roberto Paroni} } @article {2011, title = {The time-dependent von K{\'a}rm{\'a}n plate equation as a limit of 3d nonlinear elasticity}, journal = {Calculus of Variations and Partial Differential Equations 41 (2011) 241-259}, number = {SISSA;74/2009/M}, year = {2011}, publisher = {Springer}, abstract = {The asymptotic behaviour of the solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness $h$ of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of $h$, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von K\\\\\\\'arm\\\\\\\'an plate equation.}, doi = {10.1007/s00526-010-0360-0}, url = {http://hdl.handle.net/1963/3835}, author = {Helmut Abels and Maria Giovanna Mora and Stefan M{\"u}ller} } @article {2010, title = {Convergence of equilibria of thin elastic rods under physical growth conditions for the energy density}, number = {SISSA;67/2010/M}, year = {2010}, abstract = {The subject of this paper is the study of the asymptotic behaviour of the equilibrium configurations of a nonlinearly elastic thin rod, as the diameter of the cross-section tends to zero. Convergence results are established assuming physical growth conditions for the elastic energy density and suitable scalings of the applied loads, that correspond at the limit to different rod models: the constrained linear theory, the analogous of von K{\'a}rm{\'a}n plate theory for rods, and the linear theory.}, url = {http://hdl.handle.net/1963/4086}, author = {Elisa Davoli and Maria Giovanna Mora} } @article {2010, title = {Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity}, journal = {Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010) 253-295}, number = {SISSA;12/2008/M}, year = {2010}, abstract = {We discuss the limiting behavior (using the notion of gamma-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d surface. In particular, under the assumption that the elastic energy of deformations scales like h4, h being the thickness of a shell, we derive a limiting theory which is a generalization of the von Karman theory for plates.}, doi = {10.2422/2036-2145.2010.2.02}, url = {http://hdl.handle.net/1963/2601}, author = {Marta Lewicka and Maria Giovanna Mora and Mohammad Reza Pakzad} } @article {2009, title = {A nonlinear theory for shells with slowly varying thickness}, journal = {C. R. Math. 347 (2009) 211-216}, number = {SISSA;23/2008/M}, year = {2009}, abstract = {We study the Γ-limit of 3d nonlinear elasticity for shells of small, variable thickness, around an arbitrary smooth 2d surface.}, doi = {10.1016/j.crma.2008.12.017}, url = {http://hdl.handle.net/1963/2632}, author = {Marta Lewicka and Maria Giovanna Mora and Mohammad Reza Pakzad} } @article {2008, title = {Convergence of equilibria of three-dimensional thin elastic beams}, journal = {Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 873-896}, number = {SISSA;69/2006/M}, year = {2008}, abstract = {A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section tends to zero. More precisely, we show that stationary points of the nonlinear elastic functional $E^h$, whose energies (per unit cross-section) are bounded by $Ch^2$, converge to stationary points of the $\\\\varGamma$-limit of $E^h/h^2$. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James and M{\"u}ller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.}, doi = {10.1017/S0308210506001120}, url = {http://hdl.handle.net/1963/1896}, author = {Maria Giovanna Mora and Stefan M{\"u}ller} } @article {2008, title = {Globally stable quasistatic evolution in plasticity with softening}, journal = {Netw. Heterog. Media 3 (2008) 567-614}, number = {SISSA;23/2007/M}, year = {2008}, abstract = {We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each\\ntime interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.}, url = {http://hdl.handle.net/1963/1965}, author = {Gianni Dal Maso and Antonio DeSimone and Maria Giovanna Mora and Massimiliano Morini} } @article {2008, title = {A second order minimality condition for the Mumford-Shah functional}, journal = {Calc. Var. Partial Differential Equations 33 (2008) 37-74}, number = {SISSA;82/2006/M}, year = {2008}, abstract = {A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\\\\Gamma)$, $\\\\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\\\\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.}, doi = {10.1007/s00526-007-0152-3}, url = {http://hdl.handle.net/1963/1955}, author = {Filippo Cagnetti and Maria Giovanna Mora and Massimiliano Morini} } @article {2008, title = {A vanishing viscosity approach to quasistatic evolution in plasticity with softening}, journal = {Arch. Ration. Mech. Anal. 189 (2008) 469-544}, number = {SISSA;38/2006/M}, year = {2008}, abstract = {We deal with quasistatic evolution problems in plasticity with softening, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate-independent problems to the case of a nonconvex energy functional. We argue that, in this case, the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. Thus, we analize a different selection criterion for the solutions of the quasistatic evolution problem, based on a viscous approximation. This leads to a generalized formulation in terms of Young measures, developed in the first part of the paper. In the second part we apply our approach to some concrete examples.}, doi = {10.1007/s00205-008-0117-5}, url = {http://hdl.handle.net/1963/1844}, author = {Gianni Dal Maso and Antonio DeSimone and Maria Giovanna Mora and Massimiliano Morini} } @article {2007, title = {Convergence of equilibria of planar thin elastic beams}, journal = {Indiana Univ. Math. J. 56 (2007) 2413-2438}, number = {SISSA;23/2006/M}, year = {2007}, abstract = {We consider a thin elastic strip of thickness h and we show that stationary points of the nonlinear elastic energy (per unit height) whose energy is of order h^2 converge to stationary points of the Euler-Bernoulli functional. The proof uses the rigidity estimate for low-energy deformations by Friesecke, James, and Mueller (Comm. Pure Appl. Math. 2002), and a compensated compactness argument in a singular geometry. In addition, possible concentration effects are ruled out by a careful truncation argument.}, doi = {10.1512/iumj.2007.56.3023}, url = {http://hdl.handle.net/1963/1830}, author = {Maria Giovanna Mora and Stefan M{\"u}ller and Maximilian G. Schultz} } @article {2007, title = {Derivation of a rod theory for phase-transforming materials}, number = {SISSA;21/2005/M}, year = {2007}, abstract = {A rigorous derivation is given of a rod theory for a multiphase material,starting from three-dimensional nonlinear elasticity. The stored energy density is supposed to be nonnegative and to vanish exactly on a set consisting of two copies of the group of rotations SO(3). The two potential wells correspond to the two crystalline configurations preferred by the material. We find the optimal scaling of the energy in terms of the diameter of the rod and we identify the limit, as the diameter goes to zero, in the sense of Gamma-convergence.}, doi = {10.1007/s00526-006-0039-8}, url = {http://hdl.handle.net/1963/1751}, author = {Maria Giovanna Mora and Stefan M{\"u}ller} } @article {2007, title = {Time-dependent systems of generalized Young measures}, number = {SISSA;98/2005/M}, year = {2007}, abstract = {In this paper some new tools for the study of evolution problems in the framework of Young measures are introduced. A suitable notion of time-dependent system of generalized Young measures is defined, which allows to extend the classical notions of total variation and absolute continuity with respect to time, as well as the notion of time derivative. The main results are a Helly type theorem for sequences of systems of generalized Young measures and a theorem about the existence of the time derivative for systems with bounded variation with respect to time.}, url = {http://hdl.handle.net/1963/1795}, author = {Gianni Dal Maso and Antonio DeSimone and Maria Giovanna Mora and Massimiliano Morini} } @article {2006, title = {Quasistatic evolution problems for linearly elastic-perfectly plastic materials}, journal = {Arch. Ration. Mech. Anal. 180 (2006) 237-291}, number = {arXiv.org;math/0412212v1}, year = {2006}, abstract = {The problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate-independent processes. Existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation. This provides a new approximation result for the solutions of the quasistatic evolution problem, which are shown to be absolutely continuous in time. Four equivalent formulations of the problem in rate form are derived. A strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain.}, doi = {10.1007/s00205-005-0407-0}, url = {http://hdl.handle.net/1963/2129}, author = {Gianni Dal Maso and Antonio DeSimone and Maria Giovanna Mora} } @article {2002, title = {The Calibration Method for Free-Discontinuity Problems on Vector-Valued Maps}, journal = {J. Convex Anal. 9 (2002) 1-29}, number = {SISSA;61/2001/M}, year = {2002}, publisher = {Heldermann Verlag}, abstract = {The calibration method is a classical minimality criterion, which has been recently adapted to functionals with free discontinuities by Alberti, Bouchitt{\'e}, Dal Maso. In this paper we present a further generalization of this theory to functionals defined on vector-valued maps.}, url = {http://hdl.handle.net/1963/3049}, author = {Maria Giovanna Mora} } @article {2002, title = {Local calibrations for minimizers of the Mumford-Shah functional with a triple junction}, journal = {Commun. Contemp. Math. 4 (2002) 297-326}, number = {SISSA;39/2001/M}, year = {2002}, publisher = {World Scientific}, abstract = {We prove that, if u is a function satisfying all Euler conditions for the Mumford-Shah functional and the discontinuity set of u is given by three line segments meeting at the origin with equal angles, then there exists a neighbourhood U of the origin such that u is a minimizer of the Mumford-Shah functional on U with respect to its own boundary conditions on the boundary of U. The proof is obtained by using the calibration method.}, doi = {10.1142/S0219199702000646}, url = {http://hdl.handle.net/1963/3050}, author = {Maria Giovanna Mora} } @article {2001, title = {Finite Difference Approximation of Free Discontinuity Problems}, journal = {Proc. Royal Soc. Edinb. Ser. A 131 (2001), no. 3, 567-595}, number = {SISSA;14/99/M}, year = {2001}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1228}, author = {Massimo Gobbino and Maria Giovanna Mora} } @article {2001, title = {Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set}, journal = {Ann. I. H. Poincare - An., 2001, 18, 403}, number = {SISSA;21/00/M}, year = {2001}, publisher = {SISSA Library}, doi = {10.1016/S0294-1449(01)00075-0}, url = {http://hdl.handle.net/1963/1479}, author = {Maria Giovanna Mora and Massimiliano Morini} } @article {2000, title = {Functionals depending on curvatures with constraints}, journal = {Rend. Sem. Mat. Univ. Padova 104 (2000), 173--199}, number = {SISSA;85/99/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1299}, author = {Maria Giovanna Mora and Massimiliano Morini} } @article {2000, title = {Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets}, journal = {J. Math. Pures Appl. 79, 2 (2000) 141-162}, number = {SISSA;47/99/M}, year = {2000}, publisher = {SISSA Library}, doi = {10.1016/S0021-7824(99)00140-3}, url = {http://hdl.handle.net/1963/1261}, author = {Gianni Dal Maso and Maria Giovanna Mora and Massimiliano Morini} }